#### Vol. 7, No. 2, 2013

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Base change behavior of the relative canonical sheaf related to higher dimensional moduli

### Zsolt Patakfalvi

Vol. 7 (2013), No. 2, 353–378
##### Abstract

We show that the compatibility of the relative canonical sheaf with base change fails generally in families of normal varieties. Furthermore, it always fails if the general fiber of a family of pure dimension $n$ is Cohen–Macaulay and the special fiber contains a strictly ${S}_{n-1}$ point. In particular, in moduli spaces with functorial relative canonical sheaves Cohen–Macaulay schemes can not degenerate to ${S}_{n-1}$ schemes. Another, less immediate consequence is that the canonical sheaf of an ${S}_{n-1}$, ${G}_{2}$ scheme of pure dimension $n$ is not ${S}_{3}$.

##### Keywords
canonical sheaf, relative canonical sheaf, dualizing complex, relative dualizing complex, base change, depth, moduli of stable varieties
##### Mathematical Subject Classification 2010
Primary: 14J10
Secondary: 14D06, 14F10, 14E30